BICONED GRAPHS, EDGE ROOTED FORESTS, AND h-VECTORS OF MATROID COMPLEXES PRESTON CRANFORD, ANTON DOCHTERMANN, EVAN HAITHCOCK, JOSHUA MARSH, SUHO OH, AND ANNA TRUMAN ABSTRACT. A well-known conjecture of Richard Stanley posits that the h-vector of the independence complex of a matroid is a pure O-sequence. The conjecture has been estab- lished for various classes but is open for graphic matroids. A biconed graph is a graph with two specified ‘coning vertices’, such that every vertex of the graph is connected to at least one coning vertex. The class of biconed graphs includes coned graphs, Ferrers graphs, and complete multipartite graphs. We study the h-vectors of graphic matroids arising from biconed graphs, providing a combinatorial interpretation of their entries in terms of ‘edge- rooted forests’ of the underlying graph. This generalizes constructions of Kook and Lee who studied the M¨obius coinvariant (the last nonzero entry of the h-vector) of graphic ma- troids of complete bipartite graphs. We show that allowing for partially edge-rooted forests gives rise to a pure multicomplex whose face count recovers the h-vector, establishing Stan- ley’s conjecture for this class of matroids. 1. INTRODUCTION A matroid is a combinatorial structure that generalizes various notions of independence that arise in linear algebra, field extensions, graph theory, matching theory, and other areas. A graphic matroid M(G) has its ground set given by the edge set of some finite connected graph G, with independent sets given by the sets of edges that do contain a cycle. Given a matroid M, of particular interest is the number of independent sets of M of a certain size. The h-vector of M encodes this information in a convenient format. The h arXiv:2005.09138v2 [math.CO] 20 May 2020 -vector of a matroid provides topological information regarding underlying simplicial complexes and also relates to the notion of activity of bases. In his work surrounding the Upper Bound Conjecture [18], Stanley proved that if a simplicial complex is Cohen-Macaulay (an algebraic condition on its associated face ring) then its h-vector is necessarily an O-sequence: the entries hi are given by the number of degree i monomials in some order ideal (see Section 2 for details). Motivated by these results and the orderly structure of matroids (a type of Cohen-Macaulay simplicial com- plex), Stanley conjectured [18] that the h-vectors of matroids satisfy a stronger condition. Conjecture 1.1. The h-vector of a matroid is a pure O-sequence. Date: May 21, 2020. 1 2 P. CRANFORD, A. DOCHTERMANN, E. HAITHCOCK, J. MARSH, S. OH, AND A. TRUMAN Here an O-sequence is pure if the maximal elements of the underlying order ideal can be chosen to all have the same degree; again we refer to Section 2 for details. Despite re- ceiving considerable attention for over four decades, the conjecture remains mostly wide open today. It has been established for some specific classes of matroids, in particular for cographic matroids by Merino in [13], lattice-path matroids by Schweig in [17], co- transversal matroids by Oh in [16], rank 3 matroids by H´a, Stokes, and Zanello in [6], paving matroids by Merino, Noble, Ramirez-Ibanez, and Villarroel-Flores [14], and rank 4 matroids by Klee and Samper in [7]. In [11] Kook established Stanley’s conjecture for the graphic matroid of a coned graph, by definition a graph G^ = G ∗ {v} obtained from connecting a vertex v to every vertex of an arbitrary finite graph G. Kook proved the conjecture by explicitly constructing a multicomplex of ‘partially edge-rooted forests’ in G. A spanning tree T of G^ corresponds to a partially edge-rooted forest of G in such a way that the number of internally passive edges in T is given by the cardinality of edges and edge roots in its corresponding partially edge-rooted forest. In [12] Kook and Lee studied the h-vectors of complete bipartite graphs Km+1,n+1 and ⊥ provided a combinatorial interpretation for their M¨obius coinvariant µ (Km+1,n+1), which can be seen to coincide with the last nonzero entry of the h-vector of the underlying ma- troid. They showed that the set of such trees correspond to certain ‘edge-rooted forests’ of the subgraph Km,n. These constructions provide bijective combinatorial proofs for the for- ⊥ ⊥ mulas for µ (Km+1) and µ (Km+1,n+1) previously established by Novik, Postnikov, and Sturmfels in [15]. In this paper we study h-vectors of biconed graphs. By definition a biconed graph GA,B has a pair of vertices 0 and 0 such that every vertex in GA,B is adjacent to one of 0 or 0 (or both). Loops and some, but not all, parallel edges are admissible (see Definition 3.1 for a precise statement, and in particular the meaning of A and B). The class of biconed graphs includes coned graphs, complete multipartite graphs, and Ferrers graphs. In the concluding section of [12] the authors suggest biconed graphs as a class of graphs for which their ‘edge-rooted forests’ may naturally generalize. In this paper we confirm this, showing that the set of completely passive spanning trees of a biconed graph GA,B is A,B in correspondence with the collection of maximal ‘2-edge-rooted forests’ of Gred , a certain ‘reduced’ subgraph of GA,B. Furthermore, we show that by allowing for partially rooted forests this construction gives rise to a notion of ‘degree’ (in terms of the number of edge roots), in such a way that that the number of internally active edges in a spanning tree of GA,B is given by the degree in the corresponding partially 2-edge-rooted forest. Our main results can be summarized as follows. We refer to later sections for technical definitions. BICONED GRAPHS, EDGE ROOTED FORESTS, AND h-VECTORSOFMATROIDCOMPLEXES 3 Theorem 1.2 (Corollary 3.10, Lemma 4.2). Suppose GA,B is a biconed graph with h-vector =( ) A,B h h0, h1, ... , hd . Then hi is given by the number of partially 2-edge-rooted forests in Gred of degree i. ( A,B) = We let F Gred denote the set of partially 2-edge-rooted forests in G A ∪ B. The set ( A,B) F Gred has a pleasing combinatorial structure, as our next result indicates. A,B ( A,B) Theorem 1.3 (Lemma 4.3, Lemma 4.4). For any biconed graph G the set F Gred is a pure A,B multicomplex on the set of edges of Gred . From these we obtain our main result. Theorem 1.4 (Corollary 4.5). Stanley’s conjecture holds for graphic matroids of biconed graphs. The rest of the paper is organized as follows. In Section 2 we recall some basic notions from matroid theory and the study of pure O-sequences, and establish some notation. In Section 3 we describe our main objects of study and establish bijections between three sets: A,B A,B spanning trees of a biconed graph G , birooted forests in Gred , and 2-edge-rooted forests A,B in Gred . In Section 4, we prove that the set of the 2-edge-rooted forests is a pure multi- complex. Here we also prove that the pure O-sequence arising from this multicomplex is the h-vector of the (graphic matroid of the) underlying biconed graph, thus establishing Stanley’s conjecture. In Section 5, we suggest some further applications of 2-edge-rooted forests and also discuss some open questions. 2. PRELIMINARIES In this section we recall some basic definitions and set some notation. 2.1. Matroids. We first review some basic notions of matroid theory, referring to [2] for more details. For the purposes of this paper, a matroid M =(E, I) on a finite ground set E is a nonempty collection I of subsets of E satisfying (1) If A ∈ I and B ⊂ A then B ∈ I. (2) If A, B ∈ I and |A| > |B| then there exists some e ∈ A \ B such that B ∪ e ∈ I. The collection I is called the set of independent sets of the matroid, and a maximal in- dependent set (under inclusion) is called a basis. The number of elements in any (and hence every) basis of M is called the rank of the matroid. Given any matroid M =(E, I) one defines the dual matroid M∗ as the matroid with ground set E and independent sets I∗ = {E \ I : I ∈ I}. An important example of a matroid, particularly relevant for us, comes from graph theory. If G is a finite connected graph with vertex set V(G) and edge set E(G) (possibly with loops and multiple edges) one defines the graphic matroid M(G) with ground set 4 P. CRANFORD, A. DOCHTERMANN, E. HAITHCOCK, J. MARSH, S. OH, AND A. TRUMAN E = E(G) and independent sets given by acyclic collections of edges. The bases are then spanning trees of G, and hence the rank of M(G) is given by |V(G)| − 1. 2.2. Activity and h-vectors. The collection of independent sets of a matroid form a sim- plicial complex called the independence complex of M. Associated to a simplicial complex of dimension d − 1, and therefore to a matroid of rank d, is its f-vector f =(f−1, f0, ... , fd−1), where fi−1 is the number of simplices of cardinality i. The h-vector of the independence complex of M (which we will simply refer to as the h-vector of M) encodes the same information as f in a form that is more convenient, especially in algebraic contexts.